Integrand size = 46, antiderivative size = 316 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]
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Time = 0.36 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {808, 678, 674, 214} \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]
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Rule 214
Rule 674
Rule 678
Rule 808
Rubi steps \begin{align*} \text {integral}& = -\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {7}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx}{7 c e^3} \\ & = \frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {((2 c d-b e) (e f-d g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{e} \\ & = \frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^2 (e f-d g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{e} \\ & = \frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^3 (e f-d g)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e} \\ & = \frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\left (2 (2 c d-b e)^3 (e f-d g)\right ) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right ) \\ & = \frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {15 b^3 e^3 g+b^2 c e^2 (161 e f-206 d g+45 e g x)+c^3 \left (-526 d^3 g+3 e^3 x^2 (7 f+5 g x)-2 d e^2 x (56 f+33 g x)+d^2 e (511 f+157 g x)\right )+b c^2 e \left (612 d^2 g+e^2 x (77 f+45 g x)-d e (567 f+167 g x)\right )}{c (-c d+b e+c e x)^2}+\frac {105 (-2 c d+b e)^{5/2} (-e f+d g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{105 e^2 (d+e x)^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(947\) vs. \(2(286)=572\).
Time = 0.36 (sec) , antiderivative size = 948, normalized size of antiderivative = 3.00
method | result | size |
default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (15 c^{3} e^{3} g \,x^{3} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+45 b \,c^{2} e^{3} g \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-66 c^{3} d \,e^{2} g \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+21 c^{3} e^{3} f \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+105 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{3} c d \,e^{3} g -105 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{3} c \,e^{4} f -630 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c^{2} d^{2} e^{2} g +630 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} c^{2} d \,e^{3} f +1260 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{3} d^{3} e g -1260 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,c^{3} d^{2} e^{2} f -840 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{4} d^{4} g +840 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{4} d^{3} e f +45 b^{2} c \,e^{3} g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-167 b \,c^{2} d \,e^{2} g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+77 b \,c^{2} e^{3} f x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+157 c^{3} d^{2} e g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-112 c^{3} d \,e^{2} f x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+15 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b^{3} e^{3} g -206 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} c d \,e^{2} g +161 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} c \,e^{3} f +612 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} d^{2} e g -567 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} d \,e^{2} f -526 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d^{3} g +511 \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d^{2} e f \right )}{105 \sqrt {e x +d}\, \sqrt {-x c e -b e +c d}\, e^{2} c \sqrt {b e -2 c d}}\) | \(948\) |
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Time = 0.35 (sec) , antiderivative size = 949, normalized size of antiderivative = 3.00 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\left [-\frac {105 \, \sqrt {2 \, c d - b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} g\right )} x\right )} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c e^{3} x + c d e^{2}\right )}}, -\frac {2 \, {\left (105 \, \sqrt {-2 \, c d + b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} g\right )} x\right )} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}\right )}}{105 \, {\left (c e^{3} x + c d e^{2}\right )}}\right ] \]
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\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (286) = 572\).
Time = 0.36 (sec) , antiderivative size = 1074, normalized size of antiderivative = 3.40 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {105 \, {\left (8 \, c^{3} d^{3} e f - 12 \, b c^{2} d^{2} e^{2} f + 6 \, b^{2} c d e^{3} f - b^{3} e^{4} f - 8 \, c^{3} d^{4} g + 12 \, b c^{2} d^{3} e g - 6 \, b^{2} c d^{2} e^{2} g + b^{3} d e^{3} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{9} d^{2} e f - 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{8} d e^{2} f + 105 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{7} e^{3} f - 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{9} d^{3} g + 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{8} d^{2} e g - 105 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{7} d e^{2} g + 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{8} d e f - 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{7} e^{2} f - 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{8} d^{2} g + 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{7} d e g + 21 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{7} e f - 21 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{7} d g + 15 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{6} g}{c^{7}}\right )}}{105 \, e^{2}} - \frac {2 \, {\left (840 \, c^{4} d^{3} e f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 1260 \, b c^{3} d^{2} e^{2} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 630 \, b^{2} c^{2} d e^{3} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 105 \, b^{3} c e^{4} f \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 840 \, c^{4} d^{4} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 1260 \, b c^{3} d^{3} e g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) - 630 \, b^{2} c^{2} d^{2} e^{2} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 105 \, b^{3} c d e^{3} g \arctan \left (\frac {\sqrt {2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right ) + 644 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{3} d^{2} e f - 644 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c^{2} d e^{2} f + 161 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{2} c e^{3} f - 764 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} c^{3} d^{3} g + 824 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b c^{2} d^{2} e g - 251 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{2} c d e^{2} g + 15 \, \sqrt {2 \, c d - b e} \sqrt {-2 \, c d + b e} b^{3} e^{3} g\right )}}{105 \, \sqrt {-2 \, c d + b e} c e^{2}} \]
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Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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